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Using Chebyshev Polynomials to Approximate Partial Differential Equations

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  • Guglielmo Caporale

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  • Mario Cerrato

Abstract

This paper suggests a simple method based on a Chebyshev approximation at Chebyshev nodes to approximate partial differential equations. It consists in determining the value function by using a set of nodes and basis functions. We provide two examples: pricing a European option and determining the best policy for shutting down a machine. The suggested method is flexible, easy to programme and efficient. It is also applicable in other fields, providing efficient solutions to complex systems of partial differential equations.
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Suggested Citation

  • Guglielmo Caporale & Mario Cerrato, 2010. "Using Chebyshev Polynomials to Approximate Partial Differential Equations," Computational Economics, Springer;Society for Computational Economics, vol. 35(3), pages 235-244, March.
  • Handle: RePEc:kap:compec:v:35:y:2010:i:3:p:235-244
    DOI: 10.1007/s10614-009-9172-8
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    File URL: http://hdl.handle.net/10.1007/s10614-009-9172-8
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    References listed on IDEAS

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    1. Avinash Dixit, 1992. "Investment and Hysteresis," Journal of Economic Perspectives, American Economic Association, vol. 6(1), pages 107-132, Winter.
    2. Elias Tzavalis & Shijun Wang, 2003. "Pricing American Options under Stochastic Volatility: A New Method Using Chebyshev Polynomials to Approximate the Early Exercise Boundary," Working Papers 488, Queen Mary University of London, School of Economics and Finance.
    3. Abadir, Karim M. & Rockinger, Michael, 2003. "Density Functionals, With An Option-Pricing Application," Econometric Theory, Cambridge University Press, vol. 19(05), pages 778-811, October.
    4. Kenneth L. Judd, 1998. "Numerical Methods in Economics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262100711, January.
    5. Dangl, Thomas & Wirl, Franz, 2004. "Investment under uncertainty: calculating the value function when the Bellman equation cannot be solved analytically," Journal of Economic Dynamics and Control, Elsevier, vol. 28(7), pages 1437-1460, April.
    6. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
    7. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," University of California at Los Angeles, Anderson Graduate School of Management qt43n1k4jb, Anderson Graduate School of Management, UCLA.
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    Citations

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    Cited by:

    1. Niko Jaakkola, 2013. "Putting OPEC Out of Business," OxCarre Working Papers 099, Oxford Centre for the Analysis of Resource Rich Economies, University of Oxford.
    2. Alejandro MosiƱo, 2012. "Using Chebyshev Polynomials to Approximate Partial Differential Equations: A Reply," Computational Economics, Springer;Society for Computational Economics, vol. 39(1), pages 13-27, January.
    3. Polychronis Manousopoulos & Michalis Michalopoulos, 2015. "Term structure of interest rates estimation using rational Chebyshev functions," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 38(2), pages 119-146, October.

    More about this item

    Keywords

    European options; Chebyshev polynomial approximation; Chebyshev nodes; C63; G12;

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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